Journal of the Korean Operations Research and Management Science Society (한국경영과학회지)

The Korean Operations Research and Management Science Society (한국경영과학회)
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An Ant Colony Optimization Algorithm to Solve Steiner Tree Problem

스타이너 트리 문제를 위한 Ant Colony Optimization 알고리즘의 개발

  • 서민석 (펜실바니아 주립대학교 산업공학과) ;
  • 김대철 (한양대학교 경영학부)
  • Published : 2008.09.30
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Abstract

The Steiner arborescence problem is known to be NP-hard. The objective of this problem is to find a minimal Steiner tree which starts from a designated node and spans all given terminal nodes. This paper proposes a method based on a two-step procedure to solve this problem efficiently. In the first step, graph reduction rules eliminate useless nodes and arcs which do not contribute to make an optimal solution. In the second step. ant colony algorithm with use of Prim's algorithm is used to solve the Steiner arborescence problem in the reduced graph. The proposed method based on a two-step procedure is tested in the five test problems. The results show that this method finds the optimal solutions to the tested problems within 50 seconds. The algorithm can be applied to undirected Steiner tree problems with minor changes. 18 problems taken from Beasley are used to compare the performances of the proposed algorithm and Singh et al.'s algorithm. The results show that the proposed algorithm generates better solutions than the algorithm compared.

Keywords

References

  1. Bahiense L., F. Barahona and O. Porto, "Solving Steiner Tree Problems in Graphs with Lagrangian Relaxation," J. of Combinatorial Optimization, Vol.7(2003), pp.259-282. https://doi.org/10.1023/A:1027368621279
  2. Beasley, J.E., "An Algorithm for the Steiner Problem in Graphs," Networks, Vol.14(1984), pp.147-159. https://doi.org/10.1002/net.3230140112
  3. Blum, C. and M. Dorigo, "The Hyper-Cube Framework for Ant Colony Optimization," IEEE Transactions on Systems, Man, and Cybernetics- Part B, Vol.34(2004), pp.1161-1172. https://doi.org/10.1109/TSMCB.2003.821450
  4. Dorigo, M., and T. Stützle, Ant Colony Optimization., The MIT press, London, England, 2004.
  5. Dorigo, M. and L.M. Gambardella, "Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem," IEEE Transactions on Evolutionary Computation, Vol.1 (1997), pp.53-66. https://doi.org/10.1109/4235.585892
  6. Dowsland, K.A. Hill-climing, "Simulated Annealing and the Steiner Problem in Graphs," Eng. Optimization, Vol.17(1991), pp.91-107. https://doi.org/10.1080/03052159108941063
  7. Duin, C.W and S. Voss, "Steiner tre heuristics- A survey," Operations Research Proceedings, Springer-Verlag, Berlin, 1994, pp.485-496.
  8. Duin, C.W. and A, Volgenant, "Reduction Tests for the Steiner Problem in Graphs," Networks, Vol.19(1989), pp.549-567. https://doi.org/10.1002/net.3230190506
  9. Esbensen, H., "Computing Near-Optimal Solutions to the Steiner Problem in a Graph Using a Genetic Algorithm," Networks, Vol.26(1995), pp.173-185. https://doi.org/10.1002/net.3230260403
  10. Kapsalis, A., V.J. Rayward-Smith, and G.D. Smith, "Solving the Graphical Steiner Tree Problem Using Genetic Algorithms," Journal of Operation Research Society, Vol.44(1993), pp. 397-406. https://doi.org/10.1057/jors.1993.69
  11. Garey, M.R. and D.S. Johnson, Computers and Intractability : A Guide to Theory of NP-Completeness. Freeman, New York, 1979.
  12. Hwang, F.K., D.S. Richard, and p.Winter, The Steiner Tree Problem. North-Holland, Publishing Company, Amsterdam, Netherlands. 1992.
  13. Koch, T., A. Martin, and S. Voss, "SteinLib : An Updated Library on Steiner Tree Problems in Graphs," Technical Report ZIB-Report 00- 37, Konrad-Zuse-Zentrum fűr Informationstechnik Berlin, 2000.
  14. Maculan, N., p.Souza and A.C. Vejar, "An Approach for the Steiner Problem in Directed Graphs." Annals of Operations Research, Vol.33 (1991), pp.471-480. https://doi.org/10.1007/BF02071983
  15. Melkonian V. "New primal-dual algorithms for Steiner tree problems," Vol.34(2007), pp.2147-2167.
  16. Minoux, M., "Efficient Greedy Heuristics for Steiner Tree problems Using Reoptimization and Supermodularity," INFOR, Vol.28(1990), pp.221-233.
  17. Prim, R.C. "Shortest Connection Networks and Some Generalisations," Bell System Technical Journal, Vol.36(1957), pp.1389-1401. https://doi.org/10.1002/j.1538-7305.1957.tb01515.x
  18. Ribeiro, C.C. and M.C. De Souza, Tabu Search for the Steiner Problem in Graphs. Networks, Vol.36(2000), pp.138-146. https://doi.org/10.1002/1097-0037(200009)36:2<138::AID-NET9>3.0.CO;2-U
  19. Schiemangk, C. "Thermodynmically Motivated Simulation for Solving the Steiner Tree Problem and the Optimization of Interacting Path Systems," Optimization of Connection Structures in Graphs, Iwainsky, A(ed.), CICIP, East Berlin, GDR, 1985, pp.74-90.
  20. Singh, G., S. Das, S. Gosavi, and S. Pujar, Ant colony algorithms for Steiner trees: an application to routing in sensor networks. In Recent Developments in Biologically Inspired Computing, L.N. de Castro and F.J. von Zuben, Eds. Idea Group Inc., Chapter, Vol.6(2004).
  21. Voss, S. "Steiner's Problem in Graphs: Heuristic Methods," Discrete Applied Mathematics, Vol. 40(1992), pp.45-72. https://doi.org/10.1016/0166-218X(92)90021-2
  22. Xu, j., S.Y. Chiu, and F. Glover, "Using Tabu Search to Solve the Steiner Tree-Star Problem in Telecommunications Network Design, Telecommunications Syst, Vol.6(1996), pp.117-125. https://doi.org/10.1007/BF02114289
  23. Xu, J., S.Y. Chiu, and F. Glover, "Probabilistic Tabu Search for Telecommunications Network Design," Combin Optim Theory Pract, Vol.1 (1997), pp.69-94.